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Chapter 6. Light Optics

6.1 Properties of Light

6.1.1. Snell's Law (Refraction)

When a beam of light strikes a polished surface of an isotropic material
with a different index of refraction, it is split into two rays, a reflected
and a refracted ray. The reflected ray always has the same angle to the normal
as the incident beam. The refracted beam does not, but the angles which the
incident and refracted ray make with the normal to the surface are
related by the equation:

ni sin i = nr sin r

The reflected and refracted beams are partially polarized in a complimentary
fashion and maximum polarization is achieved when the angles of incidence and refraction
are complimentary (i.e. i + r = 90º) (sin r = cos i). In going from
a high index to a low index medium, there is a critical
angle above which the ray cannot escape and there is Total Internal
Reflection. For our glass block (n = 1.50):

1.5 sin i = 1.0 sin 90º
sin i = 1/1.5 sin 90º = 0.67
i = 41.8 º

An ordinary beam of light has a radially symmetry distribution of electric and
magnetic vectors in the plane normal to its path. It has a distribution of wavelengths
across the visable spectrum and the electric and magnetic vectors of the various
corpuscles are not in phase.

We can insert a color filter which only permits a certain range of wavelengths.
If this range is narrow or if the light originates from a peculiar source, it can be all
of one wavelength. Such a beam is said to be monochromatic. We can insert
a different kind of filter which only permits light vibrating in one direction to pass.
Such a filter is called a polarizer .

Light from a laser is monochromatic, coherent (all in phase) and can be polarized.
In our microscopy we will generally use polarized, polychromatic light, although
we sometimes use monochromatic light for specific applications. Light is an electromagnetic
radiation having a wavelength (lambda) and a frequency (nu) such that the wavelength
times the frequency is the velocity

c = lambda * nu

where c is the speed of light and is equal to 3x108 m/sec or 3x1018
Å/sec in a vacuum or in air; nu is the frequency in cycles per second (hertz), and
lambda is the wavelength. In a vacuum, light travels at the same speed in all directions,
but when it enters a non-opaque substance it slows down. The ratio of the speed of lighta
in a substance to the speed of light in a vacuum is called the Refractive Index of the
substance. More about this later, but first let's look at some properties of light.

Light may be treated as a wave, but it also is corpuscular in nature. It has an electric
vector (E) and a magnetic vector (H) which are normal to each other and to the path
vector (C).

Each corpuscle of light is called a photon and has energy equal to the frequency times a
constant (Planck's constant = 6.625x10-34 joules sec) so that high frequency
(short wavelength) photons are high energy, and low frequency (long wavelength) radiation
is of low energy.

6.1.3. Dispersion

The separation of a beam of light into its component colors is known as dispersion.
It occurs because the velocity (i.e. index of refraction) of the various wavelengths
of light is different. This causes the dispersion of sunlight into a spectrum of colors
by a glass prism.

These are concave-upward curves unless plotted on a specially scaled paper called Hartmann
paper, on which they plot as straight lines. The relative dispersion curves of minerals
and oils are important for precisely determining the index of refraction for a given mineral.
The index of refraction may be used to identify a mineral.

6.1.4. Absorption

Light is attenuated on entering any dense medium by an amount which is proportional to
the distance travel in the medium. The attenuation equation is known as Lambert's Law
and applies to any radiation.

I / Io = exp(-kt)

where k is the absorption coefficient in m-! and t is the distance traveled in m.
The absorption may be a function of wavelength or direction (in a non-isotropic medium).
We may plot the transmission ratio (I/Io) as a function of wavelength.

6.1.5. Color

Selective absorption as a function of wave length gives rise to colored minerals (such as
spinels). Color in minerals observed in plane-polarized light in the optical microscope (i.e.
strong color) is an indication of the presence of transition metals (principally Ti, V, Cr, Mn,
Fe) in the mineral. This is because elements with unfilled d-orbitals have electronic tran
sitions at about the same wavelengths as optical photons and so cause strong absorptions at
specific wavelengths.

Note: color observed in crossed polarizers is due to interference
and is an entirely different phenomenon.

Selective absorption as a function of both wavelength and crystallographic direction gives
rise to pleochroic minerals (such as biotite). Pleochroism may only be observed
in plane polarized light. The trioctahedral micas are strongly pleochroic and change color when
rotated in plane polarized light. This is not to be confused with interference
or birefringence which causes pronounced color changes in cross-polarized light as
noted above.

6.1.7 Summary of Properties of Light

Wavelength (lambda)

Polarization

Frequency, (n)

Monochromatic Light

Speed of light,

cPolychromatic Light

Planck's Constant, h

Coherence (lasers)

Refraction

Snell's Law:nisin i = nr sin r

Critical Angle

Dispersion

Absorption

Lambert's Law: I / Io = e(-kt)

Color

Pleochroism

6.2. The Petrographic (polarizing) Microscope

6.2.1. Grain Mounts

We will illustrate the principals of optical properties of minerals using grain mounts).
A mineral is crushed (and seived, maybe) and placed on a standard petrographic
slide. A cover slip is placed over the grains and a drop of oil of known index is
placed beside the cover slip. This oil will be drawn under the cover slip by surface tension.

CAUTION: Oils contain trichloroethylene or bromo- or iodo-ethylene
which are nasty and unpleasant. Skin contact should be avoided.

6.2.2. Relief

If a mineral grain is immersed in an oil with identical index of refraction, absorption
and dispersion, it will disappear. This will only be true for glasses or isometric
(cubic) minerals because no liquid has anisotropic optical properties. However,
although minerals and oils are both commonly colorless and isotropic, they will generally
have slightly different dispersion curves. If the mineral and oil differ substantially
in their index of refraction, the mineral is said to stand out in "relief" and a
dark line will appear to surround the mineral grain. Relief may be positive or negative.
That is, the mineral may be of higher or lower index than the oil. Positive relief
occurs when the mineral's refractive index is higher than the oil's. Negative relief
occurs when the mineral's refractive index is lower than that of the oil.

6.2.3. Becke Line Method

In lab we saw the Becke Line method of determining the index of refraction at a
wavelength of 589 nm, as well as the Oblique Illumination method. Both of these
methods utilize white light and make use of the relative dispersion of the immersion oil
and the mineral grain to match indices for the grain and oil at lD.

Note that the dispersion factor is generally higher for liquids than for solids.
This means that when plotted on Hartmann paper, the curve for liquids is steeper than for
solids. If the two curves cross dor the D wavelength (5890 Å),
then the blue colors will be in negative relief and the red in positive
relief (blue Becke line out; yellow line in). Note also that dispersion increases
sharply with increasing index of refraction. This gives rise to the Becke Line chart
(Figure 4-2 in Jones and Bloss) and means that the dispersion colors are much more
readily seen at high index. Note that oils change index rapidly with increasing
temperature.

6.3. Non-Isotropic Materials

6.3.1. Introduction

The speed of light in a substance is an inverse function of density, as we saw when we
heated up the oil. Cubic and amorphous solids have the same density in all directions,
as do liquids. These substances are said to be isotropic. However, if we deform a cubic solid
or have a non-cubic solid, light may have different indices of refraction in different
directions. Most minerals are non-opaque, non-isotropic substances. Hexagonal, trigonal,
tetragonal, orthorhombic, monoclinic, and triclinic minerals may have different indices
of refraction in different directions. However, the indices are constrained by the overall
symmetry of the lattice. The property of having different indices of refraction in
different directions is called birefringence). This property is illustrated by the calcite
rhomb. When non-polarized light enters the calcite rhomb, it is split into
two polarized rays. This gives rise to two images. One of the rays obeys Snell's Law,
the other does not.

The index of refraction of a crystal for light vibrating in a given direction may be
represented by a vector. For isotropic ma terials the index is the same in all direc
tions, and the figure described by these vectors is a sphere. All sections through a
sphere are circular. For hexagonal, trigo nal and tetragonal minerals, the figure is
not a sphere, but a spheroid. A spheroid is a sphere which has been squashed (ob
late) or extended (prolate). Such figures have only one circular section, and one
axis normal to the circular section. They are called uniaxial. Minerals that crystallize
in the hexagonal, trigonal, and tetragonal crystal systems are optically uniaxial.
The uniaxial indicatrix figure is constrained, so that the optic axis (pole to the circular
section) is always parallel to the c crystallographic axis. Therefore, any random
section through the center will be an ellipse, one of whose axis will be normal to the
optic axis (w = Omega). The other axis is called e' (Epsilon). Cubic crystals, amorphous
solids, liquids, and gases are optically isotropic. Non-cubic crystals are optically
anisotropic.

6.3.3. Wave Normals, Ray Paths, and Vibration Directions

For anisotropic media, the wave normal is normal (perpendicular) to the vibration
direction, but is not (necessarily) parallel to the ray path. When polarized
light enters an anisotropic crystal, it is broken up into two rays whose vibration directions
are parallel to the major and minor axes of the elliptical section of the indicatrix of
the crystal, which is normal to the incident ray. Therefore, if a ray enters parallel
to the c axis of the uniaxial crystal, it is not broken up at all because E' = w, and the
section is circular (just as if it were isotropic). When the rays emerge from the crystal
and go back into an isotropic medium, the light rays destructure by interference with each
other, so that the net polarization is different for the incident light and the wavelength
of the emergent light. When the analyser is inserted, you see only the difference between
the two rays that have interacted in the crystal. This appears as a play of interference
colors.

6.3.4. The Conoscope and Uniaxial Interference Figures

Setup the conoscope on the microscope

1. Condenser in

2. High powered objective in

3. Bertrand lens in or ocular removed

4. analyser in (i.e. crossed nicols)

The conoscope can be used to obtain optical information not readily available in the
orthoscope, by making use of the third dimension. The condenser causes light to
converge in the sample, and the transmitted light gives an interference pattern on
the back focal plane of the objective. You can visualize the figure by either removing
the ocular or by inserting the Bertrand lens.

The light at the center of the figure has passed through the center. The light at
the edge of the image (figure) has passed through the sample at some angle from
vertical (up to 40º). Suppose you put in a uniaxial crystal, such that you are looking
down the unique axis (c axis = optic axis). Light that comes straight
through the center will be blocked by the analyser. This gives rise to a uniaxial optic
axis figure. The limbs of the curves are called isogyres, and the center is
called the melatope. Lines of equal retardation circle the center and are called
isochromes. The melatope corresponds to the optic axis of the sample.
If the optic axis is not exactly vertical, the figure will appear off center,
and the center will precess about the center of the field of view as the stage is rotated.

The optic sign (- or +) can be readily determined from an optic axis figure by inserting
one of the accessory plates. If addition occurs in the quadrants parallel to the slow
ray of the plate, the crystal is positive. If subtraction occurs in these quadrants,
the crystal is negative.

6.3.5. Review of Uniaxial Optical Properties

Determination of the indices of refraction in uniaxial minerals

1. Determine the sign.

a. Find grain of lowest birefringence.

b. Determine the sign from the interference figure.

2. Compare grain to oil.

a. If negative (i.e. e' << w)
and the oil is higher in relief than the grain, estimate the relief,
go to a oil of lower refractive index, and repeat.

b. If positive (i.e. e' > w)
and the oil is lower in relief than the grain, estimate the relief, go to an
oil of lower refractive index, and repeat.

c. If negative and the oil is less than w,
or positive and greater w, estimate the relief.

3. Determination of e' (epsilon) and w (omega).
a. Look for a grain showing the highest colors in crossed nicols,
rotate to extinction, then clock wise 45º and insert the accessory plate. If colors
increase, the slow ray of the grain is parallel to the slow ray of the plate.
If the colors decrease, the slow ray of the grain is perpendicular to the slow ray
of the plate. If positive, epsilon is slow, and if negative, omega is slow.
Rotate epsilon parallel to the lower nicol and uncross the nicols.
Compare the grain to the oil. Rotate 90º and compare to the oil.
This should be omega.

The biaxial indicatrix is a 3-dimensional ellipsoid. All central sections are ellipses
except 2 which are circular. The major axes of the ellipse are X, Y, and Z, and
correspond to the three indices of refraction alpha, beta, and gamma ( a
<< b << g ; X << Y << Z). The planes containing the 2
principal directions are principal planes. The normals (poles) to the 2 circular
sections are the optic axes (O.A.). Because the light travelling along these axes has the same
index of refraction for a full rotation, there is no birefringence observed when looking
down one of these axes, and such a grain is extinct for a full rotation, just as when you
are looking down the axis of a uniaxial crystal.

The angle between the optic axes is called 2V), and the plane in which they lie i
is called the optic plane). This plane is always normal to beta, the intermediate
axis, which is called the optic normal). 2V is taken as the acute angle, either 2Vz)
or 2Vx). If 2Vz) is acute (i.e. < 90º), then Z is the acute bisectrix) or Bxa), X
is the obtuse bisectrix or Bxo, and the crystal is biaxial positive.
If 2Vx is acute (i.e. < 90º), X is the acute bisectrix (Bxa), Z is
the obtuse bisectrix, and the crystal is biaxial negative. If 2V = 90º, the crystal
is neither. 2V can be calculated from alpha, beta and gamma as follows:
Cos Vz = ( a / b )
[ ( g + b )( g -
b )/( g + a )( g
- a ) ] 1/2

6.3.7. Biaxial Interference Figures

A. Centered Acute bisectrix Figure (Bxa). If 2V << 60º, the two melatopes re
main in the field of view for a full rotation.
What happens when 2V approaches 0º?
B. Centered Obtuse bisectrix Figure (Bxo)

An obtuse bisectrix figure is just like an acute bisectrix figure except that the me
latopes are never in the field of view, and the isogyres leave completely. The cross
is broad and diffuse at extinction.
C. Centered Optic Normal Figure
A broad diffuse cross fills the field of view at extinction and leaves the field in
the quadrant into which the acute bisectrix has been turned.

D. Centered Optic Axis Figure
A straight bar is formed when the optic plane is parallel to the cross hairs. When
at 45º , the bar is curved. The 2V may be estimated directly from the curve in the
bar at 45º.

6.3.8. Determination of Optic Sign

Using the acute bisectrix figure, rotate 45º off the extinction position, so that
the melatopes are parallel to the insertion direction of the plate. If addition occurs
parallel to the slow ray of the plate, the crystal is biaxial positive. If subtraction
occurs parallel to the slow direction of the plate, the crystal is biaxial negative.
This also applies to the centered optic axis figure. The isogyre is always convex towards
the acute bisectrix.

6.3.9. Methods for Estimation of 2V

1. From the curvature of the centered optic axis figure

2. Kambs Method

6.3.10. Pleochroism

Pleochroism results from different absorption of light in different directions
in a crystal. It does not occur in isotropic minerals. Uniaxial minerals only have 2 optic
directions, so dichroism is a more appropriate word to use. Pleochroic minerals change
color as they are rotated in plane polarized light (not crossed polarizers!!!).

Pleochroism is a characteristic property of biotite. Trioctahedral micas (i.e biotite)
are strongly pleochroic while dioctahedral micas (i.e muscovite) are not. Sometimes
the colors do not change markedly but total absorption does

6.4. Extinction Angles

Many minerals show characteristic optical properties in relation to their form, cleavage, twin
plane or exsolution planes. In particular the angle made between extinction and a prominant
lineation is called the extinction angle, and may be used to identify or characterize
certain minerals.

Relative to a prominant lineation, extinction may be parallel, inclined or symmetrical.
In general, hexagonal, trigonal, tetragonal, and orthorhombic crystals will show
parallel extinction. Commonly, orthorhombic minerals also show symmetrical extinction.
Monoclinic and triclinic minerals usually show inclined extinction. The extinction angle
is usually taken as the acute angle measured from the cross hair parallel to the lower
polarizer.

6.5. Thin Sections

6.5.1. Introduction

Most of you will utilize the material obtained from this course in the study of rocks in
thin section. Hence, much of the emphasis of the course has been on the tools needed
to aid this.

Thin sections come in various types. Normally, they are covered and mounted in Lakeside
Balsam (Nlakeside = 1.54). Their normal thickness is 30 micrometers (maximum
color of quartz is slightly yellow). Therefore, you should be able to estimate the
thickness of the thin section by looking at the color of quartz. Special sections may
be in apoxy (N = 1.54 to 1.58) because it does not melt or soften (used for probe mounts),
or in Crystal Bond (used for TEM) because it is acetone-soluble and may be
uncovered and polished on one side or both. Normal sections cost from $3.00 to $5.00,
polished sections from $10.00 to $15.00, and double polished sections from
$15.00 to $20.00. As I said before, most sections are in Lakeside Balsalm and covered.

6.5.2. What to do when you look at a rock in thin section

1. Low Power, Plane Polarized Light

A.) Count the number of minerals present.

B.) Note the grain size of each mineral type.

C.) Opacity

D.) Color/Pleochroism

E.) Relief and Approximate Indices of Refraction (work in from the edge)

F.) Grain Habit or Shape

Idiomorphic - has own form

Hypidiomorphic - some crystal faces

Allotriomorphic - Formless interstices

acicular - needle-like, lath-like

Dendritic

Skeletal

Rounded

Subrounded

Poikilitic (matrix)

Ophitic

Equigranular

Triple-junctions

G.) Cleavage

2. Crossed nicols, orthoscope

A.) Birefringent or Isotropic ?

B.) Maximum birefringence

C.) Extinction (parallel, inclined, symmetric, or asymmetric)

D.) Twinning

3. Conoscope

A.) Sign, Biaxial or Uniaxial

B.) Estimate 2V

C.) Dispersion ?

D.) Sign of Elongation ?

6.5.3. Reflectance

Reflectance is the percent of incident light on a surface that is reflected.
It is a function of the index of refraction which, in turn, is a function of density
(i.e. packing and mean atomic number). Hence, metals like gold (density =
19 gm/cm3) and platinum (density = 20 gm/cm3) are of high reflectance.
Fe metal also has high reflectance. Sulfides are somewhat less than metals, oxides less
than sulfides, and non-opaque silicates less than oxides. Any mineral that is a good
electrical conductor is opaque.

6.5.4. Reflected Light

You saw in Lab 16 that you could not see through some of the minerals in your thin section.
Most petrographers would simply pass these off as opaques, probably oxides, and let it go
at that. With the advent of the microprobe in recent years, many petrographers
routinely prepare polished sections, and a great deal of information is now available
from a cursory optical examination in reflected light. Oxides give temperature and
oxygen fugacity (fO2) estimates. Sulfides can give temperature estimates.

Minerals with high density and mean atomic number will tend to be opaque. This includes
most oxide minerals, all of the sulfide ores, and all of the native metal
ores. A full treatment of ore microscopy is beyond the scope of this course, but I
think that you should be exposed to some of the basics. What you need is a polished
surface and a reflected light microscope. Many of the optical phenomena observed in
transmitted light are equally well observed in reflected light. The standard reference
work is The Ore Minerals and Their Intergrowths by Paul Ramdohr.

Further Reading

Bloss, F. D. (1961) An Introduction to the Methods of
Optical Crystallography. Holt Rinehart
Winston, New York 294 pp.